To see that the approximate belief MDP facing player \(i\)  is equivalent to that facing player \(j\) , we observe that the construction of [1]} depends only on the stage cost function, the observation channel, and the unique invariant distribution on the underlying state space. The stage cost and observation channel are symmetric and shared by players \(i\)  and \(j\) , so it suffices to show that the unique invariant distribution on \(\textbf {X}\) , say \(\nu _{\mathbf {\pi }}\) , is symmetric in the following sense:
\(\nu _{\mathbf {\pi }} ( {\rm swap}_{ij} ( \textbf {s}) ) = \nu _{\mathbf {\pi }} ( \textbf {s}), \quad \forall \textbf {s}\in \textbf {X}.\) 
